Skip to main content

Full text of "Treatise On Analysis Vol-Ii"

See other formats


238   XIII    INTEGRATION

(cf. Section 12,7, Problem 6). (Argue by contradiction, and assume that for some
e > 0 there exists w0 such that md£ > h» for all m J> m0 , where hi = c» 1 -f e. For
1 :g i rg m, let Bf be the ball with center xt and radius %hnm~lln, and for m <J i <; 2nw
let B( be the ball with center Xj and radius i/zfl(2/~1/n — m"1/w). Show that the 2"m
balls B( are pairwise disjoint, and calculate the measure of their union by the Euler-
MacLaurin summation formula.)

16.    Let X, Y be two locally compact spaces and A a universally measurable subset of
X x Y.

(a)    For each x e X, show that the section A(X) of A at x is a universally measurable
subset of Y. Also, for each measure ft ;> 0 on Y, the function #h-»/u,*(A(x)) is uni-
versally measurable (use the Lebesgue-Fubini theorem).

(b)    If Y is compact and if A is closed in X x Y, then the function x h-> ju,*(A(X)) is
upper semicontinuous.

(c)    Let ^ be a positive measure on Y such that the section A^GO is denumerable
for almost all y e Y (with respect to p). Show that the set N of points x e X for which
fjb*(A(x)) > 0 contains no nondenumerable compact subset. (By using Section 3.9,
Problem 4, reduce to the case where this compact set contains no isolated point, and
show that such a set is the support of a diffuse measure ^0, by using Section 13.18,
Problem 6(b) and Section 4.2, Problem 3(b).)

17.    Let /t be a bounded positive measure on X, with total mass 1. If /e -S?c k such that
j log 1 1 + £/| fl^rgO for every complex number £, then /is /^.-negligible. (Use the

formula

27T

(which is valid for all £ e C) together with the Lebesgue-Fubini theorem, to evaluate
the integral   log+|R/| dp,9 where R > 0, and deduce that this integral must be zero.)

18.   Let I denote the discrete space {0, 1, . . . , m — 1}. Let X denote the compact product
space Iz, and p, a positive measure on I with total mass 1 (so that p, is defined by the

finite sequence of masses pj = ^({y}) such that pj ^ 0 and ^pj = 1). On each factor

j=o

Xn of X let ju,n denote the measure /x, and let v be the product measure ® \in on X

n e Z

(Problem 9).

(a)    For each x = (x)neZ in X, put u(x) == (jcn4. 1)fieZ . Show that u is a homeomor-
phism of X onto X and that the measure v is invariant under u. The triple (X, v, u)
is called the Bernoulli scheme B(j?0 , • • • > Pm- 1).

(b)    Consider in particular the Bernoulli scheme B(J, i). For each x « (xn)ne z e X,
where xn — 0 or 1 for all n e Z, let/(x) denote the canonical image in X2 of the point

00                                                                   00

(y, z) e R2, where y = £ x_n2"lt"1 and z =X xn2~~". Let j9 be the normalized Haar

n=0                               n=l

measure (14.3) on the compact group T2. Show that /is continuous and that f(v) = |9;
also that the set of points 1 6 T2 such that/"1(/) does not consist of a single point is
£ -negligible. isometric